Keep it balanced
An equation is a balance: two sides with exactly the same value. Do the same thing to both sides and it stays balanced. That one rule, with zero pairs and the habit of checking, solves every equation in Grade 7 — and it is the same rule behind every equation you will solve after this year.
Why think of an equation as a balance?
Equations are for finding a number you cannot see from things you can: how many movies someone saw from what they spent, what the temperature was before it rose, how many tables seat 30 people. You know what was done to the unknown number. You want to work backwards to it.
The balance is what makes working backwards safe. If two sides are equal and you add 5 to both, they are still equal, so every line of working is true as long as the line before it was. It is also why checking works: the answer has to make the original equation balance.
The equals sign was invented for exactly this. In 1557 the Welsh mathematician Robert Recorde, tired of writing “is equal to” over and over, drew two parallel lines of the same length instead, because he could think of no two things more equal. Before that, equations were written out in words.
Where it turns up
- Making change — a cashier counting up from a $7.35 bill to the $10 you handed over is solving 7.35 + c = 10, and the change is $2.65
- Weighing a pet — a cat will not sit still on a bathroom scale, so you weigh yourself holding it: 50 kg holding it and 45 kg without means c + 45 = 50, and the cat is 5 kg
- Unit prices — a shelf tag of 4 cans for $6 is 4c = 6, so each can costs $1.50, which is how you compare it with the single can beside it
- Taxi fares — a fare with a starting charge plus a price per kilometre is an equation: $3 to start plus $2 a kilometre, ending at $27, is 2k + 3 = 27, a 12 km ride
- 1. The words, first
- 2. What the equals sign says
- 3. The same to both sides
- 4. Zero pairs clear a side
- 5. Undo in reverse order
- 6. Check by substitution
The words, first
The idea: An expression is something you work out. An equation is something you solve. The difference is the equals sign.
| Word | What it means |
|---|---|
| Expression | Numbers, variables and operations with no equals sign, such as 3x + 5. |
| Equation | Two expressions joined by an equals sign, such as 3x + 5 = 26. |
| Side | Everything to the left of the equals sign, or everything to the right of it. |
| Variable | A letter standing for an unknown number. |
| Solution | The value of the variable that makes the equation true. |
| Preservation of equality | Doing the same to both sides keeps an equation true. |
| Inverse operation | The operation that undoes another: subtracting undoes adding, dividing undoes multiplying. |
| Zero pair | +1 and −1 together, worth 0. |
| Substitute, verify | Put a number in place of the variable; check that both sides of the original equation come out equal. |
What the equals sign says
The idea: The equals sign means “has the same value as”. It does not mean “here comes the answer”.
Worked example. What goes in the box? 3 + 4 = □ + 2.
Many people write 7, because 3 + 4 is 7. But the equals sign says the two sides are worth the same. The left is worth 7, so □ + 2 must be worth 7 too, and the box is 5.
Running equals signs. Writing 5 + 3 = 8 × 2 = 16 to mean “5 + 3 is 8, then times 2 is 16” actually claims 5 + 3 = 16, which is false. Start a new line for each step, so every equals sign tells the truth.
The same to both sides
The idea: Add, subtract, multiply or divide both sides by the same number and the balance stays level. Do it to one side only and it tips.
Worked example. Solve x − 6 = −2.
- What was done to x: 6 was subtracted.
- Undo it on both sides: add 6. The left becomes x, and the right becomes −2 + 6 = 4.
- So x = 4. Verify: 4 − 6 = −2 ✓.
Multiplying works too. For x/3 = 5, multiply both sides by 3: x = 15. Verify: 15 ÷ 3 = 5 ✓.
Both sides means all of both sides. Dividing 4x + 6 = 30 by 4 means dividing the 4x, the 6 and the 30. That is awkward, which is one reason the usual order undoes the + 6 first.
Zero pairs clear a side
The idea: To get rid of +4 next to x, add −4 to both sides. The +4 and −4 form zero pairs and vanish — which is what “move it across and change the sign” really means.
Worked example with tiles. Solve x + 4 = −3.
- Set it up: on the left, an x tile and 4 positive tiles. On the right, 3 negative tiles.
- Add 4 negative tiles to both sides. On the left they make 4 zero pairs with the positives, leaving x alone. On the right there are now 3 + 4 = 7 negative tiles.
- So x = −7. Verify: −7 + 4 = −3 ✓.
Why the shortcut can mislead. “Move the 4 across and change its sign” is only a quick way of writing what the tiles did: adding −4 to both sides. Learn only the shortcut and it is easy to “move” the 3 in 3x across as −3. That is not allowed: the 3 is multiplying x, so it has to be undone by dividing.
Undo in reverse order
The idea: Undo the last thing that was done to x first. That usually means undoing adding or subtracting before multiplying or dividing.
Why reverse. Socks go on before shoes, so shoes come off first. An equation is undone the same way.
Worked example. Solve 3x + 5 = 26.
- What was done to x, in order: multiplied by 3, then 5 added.
- Undo the last step first. Subtract 5 from both sides: 3x = 21.
- Then the first step. Divide both sides by 3: x = 7.
- Verify: 3 × 7 + 5 = 21 + 5 = 26 ✓.
One step only. In x/4 = 9 the only thing done to x was dividing by 4. Multiply both sides by 4: x = 36. Verify: 36 ÷ 4 = 9 ✓.
Check by substitution
The idea: Put the answer back into the original equation. If both sides come out equal, the answer is right, however it was found.
Worked example. Someone solves 4x + 6 = 30 by dividing by 4 first, but forgets to divide the 6. They get x + 6 = 7.5, so x = 1.5.
Check it. 4 × 1.5 + 6 = 6 + 6 = 12, and 12 is not 30. The check says something went wrong before anyone marks it. Done properly: subtract 6 to get 4x = 24, then divide by 4, so x = 6. Check: 4 × 6 + 6 = 30 ✓.
Check the original, not a later line. If a line was copied wrongly, the lines after it will happily agree with a wrong answer.
In a word problem, check against the story. If Ella paid $8 to join a movie club and $6 a movie, and your answer says she saw 6 movies, work out her bill: 6 × 6 + 8 = 44. That catches an equation set up wrongly in the first place, which substituting into the same equation never can.